Do Weapon Specialists not get the usual (Las or Solid Projectile and (Low-Tech) weapon trainings at character creation? Seems weird that they are the only specialty that can not be trained with their lasgun at char creation?

They get it, but in a fashion that let's them chose. 'Weapon Training (choose 3 excluding Heavy and Exotic)'. So they get any three weapon groups of their choice. They're not locked into Las or Solid Projectile. Now...... that said, you can sorta screw yourself and not pick up Weapon Training (Any of the weapons you carry) and be awful at your starting gear..... but so be it.

Of course if you actually did something as stupid as not being proficient with any of your weapons, you would need to follow page 101 of your Infantryman Primer: Immediately report to your local Commissar for Summary Execution!

But one might hope that common sense, and a desire to live, should make this unnecessary .